where
the derivative of F(x) is f(x). Our problem is to find F(x) when we are given
f(x).

We
know that the symbol
... dx is the inverse of or when dealing with differentials, the
operator symbols d and are the inverse of each other; that
is,

and
when the derivative of each side is taken, d annulling
, we
have

or
where annuals
, we
have

From
this, we find that

so
that,

Also
we find that

so that,

Again, we find that

so that,

This is to say that

and

where C is any constant of integration.

A number that is independent of the variable of
integration is called a constant of integration. Since C may
have infinitely many values, then a differential expression may have infinitely
many integrals differing only by the constant. This is to say that two
integrals of the same function may differ by the constant of integration. We
assume the differential expression has at least one integral. Because the
integral contains C and C is indefinite, we call

an indefinite integral of f(x) dx.
In the general form we say

With regard to the constant of integration, a theorem and
its converse are stated as follows:

Theorem 1. If two functions differ by a constant, they have the same
derivative.

Theorem 2. If two functions have the same derivative, their
difference is a constant.